Dwarf Nova Outbursts with Magnetorotational Turbulence

Mon. Not. R. Astron. Soc. 000, 1–18 (2015) Printed 11 July 2021 (MN LATEX style file v2.2)

Dwarf Nova Outbursts with Magnetorotational Turbulence

M. S. B. Coleman1?, I. Kotko2†, O. Blaes1, J.-P. Lasota2,3, and S. Hirose41 Department of Physics, University of California, Santa Barbara, CA 93106, USA2 Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences, Bartycka 18, 00-716 Warszawa, Poland3 Institut d’Astrophysique de Paris, CNRS et Sorbonne Universites, UPMC Univ Paris 06, UMR 7095, 98bis Bd Arago, 75014 Paris, France4 Department of Mathematical Science and Advanced Technology, Japan Agency for Marine-Earth Science and Technology, Yokohama,

Kanagawa 236-0001, Japan

Accepted —. Received —; in original form —

ABSTRACTThe phenomenological Disc Instability Model has been successful in reproducing theobserved light curves of dwarf nova outbursts by invoking an enhanced Shakura-Sunyaev α parameter ∼ 0.1 − 0.2 in outburst compared to a low value ∼ 0.01 inquiescence. Recent thermodynamically consistent simulations of magnetorotational(MRI) turbulence with appropriate opacities and equation of state for dwarf novaaccretion discs have found that thermal convection enhances α in discs in outburst,but only near the hydrogen ionization transition. At higher temperatures, convectionno longer exists and α returns to the low value comparable to that in quiescence. Inorder to check whether this enhancement near the hydrogen ionization transition issufficient to reproduce observed light curves, we incorporate this MRI-based variationin α into the Disc Instability Model, as well as simulation-based models of turbulentdissipation and convective transport. These MRI-based models can successfully re-produce observed outburst and quiescence durations, as well as outburst amplitudes,albeit with different parameters from the standard Disc Instability Models. The MRI-based model lightcurves exhibit reflares in the decay from outburst, which are notgenerally observed in dwarf novae. However, we highlight the problematic aspects ofthe quiescence physics in the Disc Instability Model and MRI simulations that areresponsible for this behavior.

Key words: accretion, accretion discs — MHD — turbulence — stars: dwarf novae.

1 INTRODUCTION

Dwarf novae are transient optical outbursts observed fromclose binary systems containing an accreting white dwarf.These outbursts have amplitudes up to ∼ 8 mag and last2-20 days, with recurrence times ranging from ∼ 4 daysto years (Lasota 2001). The Disc Instability Model (here-after DIM) is a well-tested model that attributes the out-bursts to a thermal-viscous1 instability that arises at tem-peratures where hydrogen is ionizing in the accretion disc.During quiescence, a cold dwarf nova disc accumulates mat-ter and heats until somewhere the temperature crosses a crit-ical value which triggers the thermal instability. This createsheating fronts which propagate into the low-temperaturezones, leaving behind ionized regions. In these hot regions of

? E-mail: [email protected]† Now at the Pennsylvania State University, 222A Computer

Building, University Park, PA 16802, USA1 To be clear, “viscosity” here and throughout the paper is an

effective turbulent viscosity as opposed to a true molecular vis-

cosity.

the disc, the turbulence and the Shakura & Sunyaev (1973)α parameter are assumed to be enhanced, leading to an in-crease in angular momentum transport. This causes mass,which during quiescence had gathered mostly in the outerparts of the disc, to diffuse inwards at a high rate. The DIMis also successful in explaining soft X-ray transient outburstsobserved in close binary systems containing accreting neu-tron stars and black holes, although X-ray irradiation of theouter disc modifies the stability criterion (van Paradijs 1996)and plays an important role in extending the duration of theoutburst in these systems (King & Ritter 1998; Dubus et al.2001).

In addition to their intrinsic interest, the observedamplitudes and time scales present in dwarf nova lightcurves provide the best quantitative measurements of thestresses responsible for angular momentum transport in ac-cretion discs (King et al. 2007). It was realized early on(e.g. Mineshige & Osaki 1983; Smak 1984; Meyer & Meyer-Hofmeister 1984) that the observed outburst amplitudes canonly be reproduced in the DIM if the stress parameter αtakes on different values in the outburst (denoted by h for

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hot) and quiescent (denoted by c for cold) states, with αh be-ing larger than αc by about a factor of ten. Modern versionsof the DIM use an interpolation scheme between constantvalues of αh and αc in the outburst and quiescent states,but this does not imply that α is necessarily constant forall high or for all low temperatures2. Detailed fits of theDIM to the observed correlations between orbital period(or, equivalently, outer disc radius) and the outburst du-ration and decay rate in normal, U Gem-type systems andnormal outbursts of SU UMa systems, give values for αh

of between 0.1 and 0.2, and certainly rule out significantlysmaller values (Smak 1999; Kotko & Lasota 2012). More-over, only such high values are able to explain the observedlinear relationship between outburst amplitude and the log-arithm of outburst recurrence time (the Kukarkin-Parenagorelation; Kotko & Lasota 2012).

Ever since the first application of the magnetorotationalinstability (MRI) to accretion discs (Balbus & Hawley 1991;Hawley & Balbus 1991), it has been widely suspected thatboth angular momentum transport and dissipation of me-chanical energy is mediated by MRI turbulence, at least ifthe disc is sufficiently electrically conducting. The accretionstress is due to correlations between radial and azimuthalmagnetic field fluctuations, as well as radial and azimuthalvelocity fluctuations (e.g. Balbus & Hawley 1998). One canmeasure this stress directly from numerical simulations ofthe turbulence. Equivalent values of the α parameter canthen be derived by scaling this stress with the average ther-mal pressure.

Because the values of α are among the most fundamen-tal ways of confronting MRI turbulence with observations,many groups have measured these values from their simu-lations. In the absence of net vertical magnetic flux, localshearing box simulations that incorporate vertical gravitygenerally give time-averaged α values of around two or threepercent (Hirose et al. 2006; Davis et al. 2010; Shi et al. 2010;Guan & Gammie 2011; Simon et al. 2012). Global simula-tions without net vertical flux of the entire disc, but whichallow for large scale field loops that can produce local ver-tical magnetic fluxes threading the disc, also so far producecomparably small values of α (Sorathia et al. 2010; Hawleyet al. 2011; Sorathia et al. 2012), though more work needs tobe done in exploring the effects of various field topologies.

A number of suggestions have been made as to how toresolve the discrepancy between the high values of α inferredfrom dwarf novae in outburst, and these low values measuredin MRI simulations. First, it has been known for some timethat shearing box simulations with no vertical gravity havestronger α values when the box is threaded by a verticalmagnetic field, and the resulting α increases with magneticfield strength so long as the critical vertical wavelength ofthe MRI lies within the box (Hawley et al. 1995; Sano et al.2004; Pessah et al. 2007).3 Because the initial total verticalmagnetic flux is necessarily conserved in shearing box simu-lations, this would appear to require imposing a net external

2 E.g. Mineshige & Wheeler (1989) used α ∼ (H/R)1.5.3 Recent work by Shi et al. (2016) has also shown that the valueof α in shearing boxes with no vertical magnetic field or vertical

gravity is sensitive to the height of the box: taller boxes produce

significantly higher values of α.

vertical magnetic field in the disc from the outside in orderto increase the value of α in outburst states. This might re-sult from the magnetic field of the companion star, althoughone would then have to explain why the resulting value of αis so universal in the outburst state.

On the other hand, the increase in local stress with localvertical magnetic flux has been confirmed in global simula-tions with no overall net vertical magnetic flux, althoughthe global disc still has a low value of α when averaged overthe entire disc (Sorathia et al. 2010). Having net vertical fluxmay also drive magnetocentrifugal winds, which can also ex-tract angular momentum from the disc (Suzuki & Inutsuka2009; Fromang et al. 2013; Lesur et al. 2013; Bai & Stone2013). In addition to the vertical magnetic field, transientphases such as those caused by the dwarf nova outburststhemselves may also produce periods of enhanced α that aresimilar to that observed in transient magnetic field growthphases in global simulations (Sorathia et al. 2012).

Even without vertical magnetic field or transient behav-ior, however, radiation MHD stratified shearing box sim-ulations that incorporate a realistic equation of state andopacities near the hydrogen ionization transition reproducethermal equilibria resembling the stable upper and lowerbranches of the local “S-curve” of the DIM (Hirose et al.2014). As in previous MRI simulations, α values of a fewpercent generally result, but α dramatically increases as oneapproaches the lower end of the upper branch. This appearsto be due to the onset of intermittent vertical convectioncaused by the large increase in opacity near the ionizationtransition. The resulting vertical motions at the beginningof the convective episodes build up vertical magnetic field,which may be seeding the axisymmetric MRI. In addition,temporal phase differences between the variations of stressand pressure may also be increasing the time-averaged α.4

These simulations therefore reproduce the observed high val-ues of α on the upper branch of the S-curve, but only nearthe low-temperature end of the upper branch.

In addition to the variations of α on the upper branch,there are significant differences in the time-averaged ver-tical structure observed in the simulations and the stan-dard assumptions used in the DIM. First, the DIM gener-ally assumes that the stress to pressure ratio is constantwith height, so that the dissipation rate per unit volumeis proportional to pressure. MRI simulations, on the otherhand, generally produce a more extended vertical dissipa-tion profile (Turner 2004; Hirose et al. 2006). Moreover, thesimulated upper, near-photosphere layers are generally sup-ported against the vertical tidal gravity by magnetic forces,not thermal pressure forces (Miller & Stone 2000; Hiroseet al. 2006), in contrast to the DIM vertical structure mod-els. Finally, as we have just discussed, vertical transport ofheat in the simulations is caused by alternating episodes ofradiative diffusion and thermal convection, at least near theend of the upper branch. The DIM incorporates the pos-sibility of convection using mixing length theory, but it isnot clear that this prescription adequately describes what ishappening in the MRI simulations.

The primary purpose of this paper is to incorporate the

4 Persistent convection can also enhance α, provided the Machnumbers of the convective motions exceed ∼ 0.01 (Hirose 2015).

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Dwarf Nova Outbursts with MRI Turbulence 3

variation of α measured in the MRI simulations into theDIM, to see whether an enhancement of α just near theend of the upper branch is sufficient to reproduce the ob-served amplitudes and time scales in dwarf nova outburstlight curves. A secondary objective is to use the dissipationand heat transport observed in the simulations to producemore physics-based vertical structure models that can be in-corporated in the DIM. In section 2, we discuss the behaviorof the α parameter in the MRI simulations, including newsimulations done at different radii in the disc from thoseoriginally done by Hirose et al. (2014). It turns out that αcan be reasonably fit by a function of just local disc effectivetemperature. In section 3 we present time-averaged verticaldissipation profiles from the simulations, and show how thesecan be incorporated into the DIM vertical model equations.We also briefly discuss the evolution of MRI simulationswhen they are not in thermal equilibrium, and show howthis evolution is surprisingly consistent with the standardeffective α prescription currently used in the DIM. Finally,we also present a mixing length prescription which appearsable to reproduce the time-averaged vertical structures ob-served in the simulations on the upper branch. In section 4,we present theoretical dwarf nova outburst light curves us-ing these MRI simulation-based prescriptions, and comparethem to the standard DIM light curves. We discuss our re-sults in section 5, and summarize our conclusions in section6.

2 MRI SIMULATION-BASED STRESSPRESCRIPTION

As discussed in the introduction, the ratio of vertically av-eraged stress to vertically averaged thermal pressure, α, canbe inferred from observations and can be measured directlyfrom simulations. The difference in α between the lower andupper branches of the S-curve has been crucial for gettingDIMs to produce realistic light curves. In the standard ver-sion of the DIM two values of alpha were used; αh ∼ 0.1 forthe hot, ionized upper branch, and αc, 4 to 10 times smaller,for the lower branch, with some smooth and rapid transitionbetween them. In particular, our modeling of the standardDIM here in this paper will use a slightly modified version ofthe prescription first introduced by Hameury et al. (1998):

log(α) = log(αc) + [log(αh)− log(αc)]

[1 +

(Tc0

Tc

)16]−1

,

(1)where Tc is the central (midplane) temperature and Tc0 is acritical temperature. The choice of this parameter is some-what arbitrary and we take Tc0 = 2.9× 104 K (Lasota et al.2008), and take the limiting α-parameters to be αc = 0.03and αh = 0.12.

Until recently, local shearing box MRI simulations(without net vertical flux through the simulation domain)produced values of α which are too low for the hot ion-ized outburst state of dwarf novae. However, for the firsttime, α measured in local MRI simulations (Hirose et al.2014) appear to be consistent with α values inferred fromobservations and commonly used to reproduce dwarf novaeoutbursts by the DIM. Hirose et al. (2014) found that theα parameter varies along the S-curve and is not constant on

the upper branch, with an enhancement of α towards the tip(low temperature end) of the upper branch. This providesan intriguing replacement for the previous ad hoc bimodalα prescription previously used (e.g. Hameury et al. 1998;Lasota 2001; Kotko & Lasota 2012).

The Hirose et al. (2014) simulations are done in thegeometry of a vertically stratified shearing box, in which asmall local patch of an accretion disc is approximated as aco-rotating Cartesian frame (x, y, z) with linearized Keple-rian shear flow −(3/2)Ωx, where x, y, and z correspond tothe radial, azimuthal, and vertical coordinates, respectively(Hawley et al. 1995). The simulations assume no explicitshear viscosity in the basic equations, and no Ohmic, Hall,or ambipolar diffusion effects are included either, i.e. idealMHD is assumed. The simulations are therefore probablynot accurate for the quiescent, largely electrically neutralstate where Ohmic dissipation and the Hall effect are likelyimportant (see Appendix A). Magnetic and kinetic energylosses at the grid scale are captured and added to the localinternal energy of the gas, creating an effective turbulentdissipation (Turner et al. 2003; Hirose et al. 2006). Addi-tionally, the simulations include realistic equation of stateand opacity tables in order to accurately model the hydro-gen ionization regime (see Hirose et al. 2014, for details), andthe flux limited diffusion approximation (which breaks downat low optical depths) is used to model radiation transportand cooling.

The simulations in Hirose et al. (2014) were computedfor only one angular frequency Ω = 6.4×10−3 s−1, which cor-responds to a distance5 of 1.25×1010 cm from a white dwarfof 0.6M. In order to explore a larger parameter space, weutilize the same methods to run simulations at two addi-tional orbital radii, 1.25×109 cm and 4.13×109 cm. Theparameters for these additional simulations can be found inTable B1.

We incorporate simulation data into the DIM’s verti-cal thermal equilibrium equations (Hameury et al. 1998) byfirst discarding the initial 10 orbits of data (this is the timeit take for turbulence to develop and erase details of the ini-tial conditions) and taking a time average of the remainingduration of the simulations. All the simulations have beenrun long enough so that at least 100 orbits (and up to 220orbits) of data are included in the time averaging. This en-sures that averaging is done over many thermal times so thatthermal fluctuations are smoothed out and that the averageis well behaved (i.e. running the simulation for another 10orbits would have little effect on the time average). We alsohorizontally average the data over the radial (x) and az-imuthal (y) directions. Finally, we vertically symmetrize thedata about the midplane z = 0. Simulation data which haveundergone these operations will henceforth be referred to asprofiles. To summarize, for some scalar data f the profile iscalculated as follows

f(z) ≡∫∫∫

[f(x, y, z, t) + f(x, y,−z, t)] dxdy dt

2∫∫∫

dxdy dt. (2)

We then fit some of these profiles in order to produce verticalstructures that more accurately represent simulation physicsthan the standard DIM.

5 Due to some rounding differences our distance slightly differs

from the 1.23×1010 cm that Hirose et al. (2014) state.

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The parameter α is computed from each simulation asfollows

α ≡∫wxy(z) dz∫

[Pgas(z) + Prad(z)] dz, (3)

where wxy is the sum of the Maxwell and Reynolds stresses

wxy ≡ −BxBy

4π+ ρvxδvy. (4)

Here Bx and By are the radial and azimuthal components ofthe magnetic field respectively6, vx is the radial velocity, andδvy ≡ vy+3Ωx/2 is the difference between the azimuthal ve-locity and the mean rotational velocity in the disc. We alsoexamined how the temporal mean of α varies with time andwe found these fluctuations diminished with time, indicat-ing that the time averaged α is well behaved. Provided thetime average is done over an interval of at least 100 orbits,differences in the time-averaged value of α within a singlesimulation are less than ten percent.

We fit the variation of α between all the simulationsat radius r = 1.25× 1010 cm with a prescription dependingonly on local effective temperature Teff of the following form

α = a exp

(−x

2

2

)+b

2tanh (x) + c, (5)

x =Teff − T0

σ. (6)

The best fit parameter values are a = 8.79×10−2, b = 2.41×10−3, c = 3.27× 10−2, T0 = 7034 K, and σ = 1000 K.Figure 1 compares this fit to the simulation data at thisradius. Also shown are the results from simulations at twoadditional radii (which are not included in the fit), whichare also reasonably consistent with this fit, though there issome indication that the simulations at smaller radii haveslightly greater convective enhancements of α. We will usethe fit of equations (5) and (6) in the simulation-based lightcurve modeling below.

3 MRI SIMULATION-BASED VERTICALSTRUCTURE MODELS

In addition to the overall behavior of the stress to pressureratio α discussed in the previous section, MRI simulationsalso exhibit differences with the standard DIM assumptionsconcerning the local vertical structure of the disc. Here weutilize data from the Hirose et al. (2014) vertically strati-fied shearing box MHD simulations to show how to makeDIM vertical structure models that better reflect some ofthe actual properties of the turbulence observed in the sim-ulations.

3.1 Dissipation Profile

Previously the DIM has relied on an ad hoc assumed verticalprofile of turbulent dissipation, in particular a vertically localα prescription in which the dissipation rate per unit volume

6 Note that the magnetic fields as defined here differ by a factor

of√

4π from the magnetic fields defined in the simulations. Here

we use standard cgs Gaussian units.

0 5 10 15 20 25

Teff / 103 K

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

α

Fitr9 = 12.5

r9 = 4.13

r9 = 1.25

Figure 1. Time averaged α versus effective temperature for all

the MRI simulations at three different radii around a 0.6M white

dwarf: r = 1.25×1010 cm (blue crosses), r = 4.13×109 cm (ma-genta crosses), and r = 1.25×109 cm (green crosses). The best fit

curve for the r = 1.25×1010 cm simulations is plotted in black.

is proportional to the local thermal pressure. This is very dif-ferent from what is observed in vertically stratified MRI sim-ulations. Such simulations of course cannot spatially resolvethe actual microscopic viscous and resistive length scales inthe plasma, and the simulations of Hirose et al. (2014) donot include any explicit resistivity or shear viscosity in thebasic equations. Instead, a total energy scheme is employedin which grid scale losses of magnetic and kinetic energy areautomatically transferred to internal energy of the plasma,thereby effecting a dissipation of turbulent mechanical en-ergy. This will capture the true dissipation rate in the tur-bulence provided the turbulent cascade that would exist inreality below the grid scale is capable of transferring most ofthe energy down to the true microscopic dissipation scales.Whether this accurately captures the dissipation occurringin real discs in nature remains to be seen.

In fact, there is strong numerical evidence that the satu-ration level of the turbulent stresses, and even whether long-lived turbulence can be maintained, depends on the valuesof the fluid and magnetic Reynolds numbers or their ratio,the magnetic Prandtl number. This is particularly true ofshearing box simulations that lack vertical gravity (e.g. Fro-mang et al. 2007; Lesur & Longaretti 2007). Including verti-cal gravity appears to slightly extend the range of magneticPrandtl numbers that allow sustained turbulence to lower(but still greather than unity) values (Davis et al. 2010).This is also true of shearing box simulations which lack ver-tical gravity, provided the box height is large enough (Shiet al. 2016). Because the simulations used here have no ex-plicit viscosity or resistivity, and dissipation effectively oc-curs at the grid scale, the effective magnetic Prandtl numbermust be of order unity. While the simulations neverthelessexhibit sustained turbulence, more work needs to be done

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to investigate whether and how the dissipation profiles, andeven the saturation level of the turbulence, might be affectedby the actual dissipation scales. Dependencies of the turbu-lent stresses on magnetic Prandtl number may even them-selves lead to thermal instabilities in accretion discs (Potter& Balbus 2014).

In any case, as shown in Figure 2, the time andhorizontally-averaged vertical profile of dissipation rate perunit volume in the simulations is not proportional to thepressure, in contrast to the simple assumption used in theDIM. In fact, the vertical dissipation profile generally doesnot even decline monotonically away from the midplane, butinstead peaks off the midplane, possibly due to the effectsof magnetic buoyancy (Blaes et al. 2011). Despite this non-monotonic behavior, we find that we can adequately replacethe usual DIM assumption of dissipation rate per unit vol-ume Q+ being simply proportional to thermal pressure with,instead, a power law dependence on thermal pressure (seeFigure 2):

Q+

Q+0

=

(P

P0

)δ, (7)

where the subscript zero is used to denote midplane values.By thermal pressure P , we mean the sum of gas pressure(Pgas) and the much smaller radiation pressure (Prad Pgas), but we exclude magnetic pressure, which typicallydominates the thermal pressure far from the midplane. Us-ing linear regression (in log space) we determined that thebest fit exponent δ = 0.35. The ratio Q+

0 /Pδ

0 is chosen suchthat the vertically integrated Shakura & Sunyaev (1973) αrelation holds:

3

2αΩ

∫ ∞−∞

P (z) dz =

∫ ∞−∞

Q+(z) dz. (8)

3.2 Non-Equilibrium Dissipation

During the passage of a heating or cooling front through thedisc, annuli at the location of the fronts are of course outof thermal equilibrium. Within the DIM (Hameury et al.1998), this is handled by constructing hydrostatic verticalstructures with a local vertical energy flux divergence givenby

dFzdz

=3

2αeffΩP (z), (9)

where P (z) is the local thermal pressure and αeff is a pa-rameter that differs from α when the annulus is not in ther-mal equilibrium, both because vertically integrated heatingand cooling will then no longer be equal and because of theconcomitant vertical thermal expansion or contraction. Theactual value that αeff takes is determined by solving for thecomplete vertical structure for a given effective temperatureand surface mass density. If thermal equilibrium does hold,then αeff = α, and we recover the standard DIM assumptionthat the dissipation rate per unit volume is proportional tothe local thermal pressure at every height in the annulus.

As just noted, the time and horizontally averaged ver-tical profiles of dissipation rate per unit volume measuredin the MRI simulations do not simply scale with local ther-mal pressure. We therefore modify equation (9) to be con-sistent with our fit to the equilibrium dissipation profile,

10−5 10−4 10−3 10−2 10−1 100 101

P/P0

10−2

10−1

100

101

Q+/Q

+ 0

Fitr9 = 12.5

r9 = 4.13

r9 = 1.25

Figure 2. Time and horizontally averaged profiles of dissipa-

tion rate per unit volume as a function of thermal pressure, eachscaled by their respective midplane values, for each of the MRI

simulations used in this paper. The different colors refer to simu-

lations done at three different radii around a 0.6M white dwarf:r = 1.25×1010 cm (blue points), r = 4.13×109 cm (magenta

points), and r = 1.25×109 cm (green points). A power law fitto all the profiles is also plotted (red line). Note that the slope

of the fit is significantly more important than the vertical off-

set, which we determine in our light curve modeling by enforcingenergy conservation using equation (8).

equation (7):

dFzdz

=3

2αeffΩP0

(P (z)

P0

)δ. (10)

However, it is not obvious that the MRI simulations shouldbehave according to this equation outside thermal equilib-rium.

If the assumption of equation (10) were perfect, thenall simulation data would fall directly on the best-fit line inFigure 2. While time averaged profiles lie near this line, itis not apparent that one should expect this behavior from athermally evolving simulation. To test this we defined

αeff(z, t) ≡ 2

3

1

ΩP0

∂Fz∂z

(P0

P

)δ, (11)

with δ = 0.35 as previously discussed, and examined vari-ations in αeff for a few simulations (see Figures 3-5). Inaddition to stable simulations (e.g. ws0446 shown in Fig-ure 3), we specifically examined two non-equilibrium simu-lations: one heating (ws0467, see Figure 4), and one cooling(ws0488, see Figure 5). These simulations were started justbeyond the edges of the lower and upper branches of theS-curve, respectively (see Figure 11 of Hirose et al. 2014).As these two simulations evolve, they move around on theplane of Teff vs. total column mass density Σ, which allowsαeff to vary with time. However, αeff should be approxi-mately constant in height at a given time if equation (10)

c© 2015 RAS, MNRAS 000, 1–18


0 20 40 60 80 100 120 140 160

Time (Orbits)

−2

−1

0

1

2

Hei

ght

/9.

51×

108

cm

ws0446

0.00 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 0.27 0.30αeff

Figure 3. Horizontally averaged αeff for simulation ws0446, a

stable convective simulation. The horizontal axis is time in orbits,and the vertical axis is height. The white dashed contours are

the photospheres. The data has been smoothed with a two orbit

running boxcar.

0 20 40 60 80 100 120 140 160

Time (Orbits)

−4

−3

−2

−1

0

1

2

3

4

Hei

ght

/1.

24×

108

cm

ws0467

0.00 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 0.27 0.30αeff

Figure 4. Horizontally averaged αeff for simulation ws0467, anunstable simulation that undergoes runaway heating until signif-

icant mass loss occurs through the vertical boundaries. The hor-

izontal axis is time in orbits, and the vertical axis is height. Thewhite dashed contours are the photospheres. The data has alsoundergone two orbit boxcar smoothing. This simulation exhibits

continuous convective vertical transport of heat.

is to adequately describe the simulation behavior. With theexception of variations near and outside the photospheres,this seems to be a reasonable approximation for simulationsws0446 and ws0467. There are some clear issues for the cool-ing simulation ws0488 just below the photosphere (see Fig-ure 5), which manifest as asymmetric regions of enhancedαeff . It is possible that these regions arise from asymmetriccooling/collapsing of the disc, which is not possible to in-corporate into the DIM and show a clear limitation of ourstrategy. However, outside these regions, ws0446 and ws0488show comparable variations, signifying that this approach isnot unreasonable.

0 10 20 30 40

Time (Orbits)

−2

−1

0

1

2

Hei

ght

/8.

12×

108

cm

ws0488

0.00 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 0.27 0.30αeff

Figure 5. Horizontally averaged αeff for simulation ws0488, an

unstable simulation that undergoes runaway cooling. The hori-zontal axis is time in orbits, and the vertical axis is height. The

white dashed contours are the photospheres. The data has also

undergone two orbit boxcar smoothing.

3.3 Mixing Length Theory

The vertical temperature profile is computed in the DIMaccording to

∇ ≡ d lnT

d lnP=

∇rad if ∇rad 6 ∇ad

∇conv if ∇rad > ∇ad

(12)

∇rad ≡3κρHpFtot

16σT 4, (13)

where∇rad and∇ad are the standard radiative and adiabatictemperature gradients, respectively, ∇conv is the convectivetemperature gradient computed using mixing length theory(Paczynski 1969; Hameury et al. 1998), κ is the Rosselandmean opacity, HP is the pressure scale height, and Ftot isthe total flux passing through a given height. Through trialand error we determined that using a value of six for themixing length parameter (αml = 6) produced vertical struc-ture models which closely resemble the time and horizontallyaveraged vertical profiles measured in the MRI simulationsthat exhibit convection (see Figure 6). This is compared tothe much lower and more conventional value of 1.5 used inthe DIM by Hameury et al. (1998). We note that if the mix-ing length theory is to be taken at face value then αml = 6implies that the length scale of convective eddies is several(∼ 6) times larger than the pressure scale height of the disc,which seems unphysical.

However, it is important to note that this value of themixing length parameter actually reflects the fact that whenconvection occurs in the simulations, it does so intermit-tently. This intermittency is the result of a limit cycle, op-erating on timescales of ∼ 10 thermal times, which is drivenby the interplay of temperature dependent opacities and en-hancement of stress by convective turbulence (see Section3.4 of Hirose et al. 2014, for further discussion). Averagingover this time dependent cycle results in a high effective αml,but the time dependent αml tend to have more canonicalvalues of ∼ 1 when convection is occurring. By measuringthe horizontally-averaged convective heat flux Fconv directlyfrom the simulations, we compute the mixing length param-eter αml that would produce this flux as a function of heightand time. We accomplished this by solving the following

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0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

T/

104

K

Simulation

DIM

TEff,Sim

0

1

2

3

4

5

6

7

P/

105

erg

cm−

3

0 5 10 15 20

z / 108 cm

0.00.20.40.60.81.01.21.41.6

ρ/

10−

7g

cm−

3

Figure 6. Comparison of vertical structure model to a con-

vective simulation (ws0441) with r = 1.25×1010 cm. Profiles oftemperature (top), thermal pressure (middle), and mass density

(bottom) are plotted verses z for profiles measured from the simu-

lation (blue) and vertical structure models (green). In these plots,it is clear that the simulations extend further than our models.

This is because our DIM vertical structures use the photosphere

as a vertical boundary condition and thus terminate there. Thusthe minimum temperature in the vertical structure model is also

the effective temperature. The time averaged effective tempera-

ture of the simulation is plotted as the dashed gray horizontalline. Thus this figure clearly shows good agreement between our

modified DIM and the simulations.

equations using the Newton-Raphson method:

α2mlβ

3/2 =2Fconv

CP ρuT, (14)

where

u ≡

√−gzHP

8

(∂ ln ρ

∂ lnT

)P

, (15)

HP ≡ −sign(z)∂z

∂ lnP, (16)

β ≡ (∇−∇′) is the positive root of the quadratic

β = (γ0αu)2 (∇−∇ad − β)2 , (17)

and

γ0 ≡CP ρ

8σT 3θ, θ ≡ 3τ

3 + τ2, τ ≡ αmlκρHP . (18)

0 20 40 60 80 100 120 140 160

Time (Orbits)

−2

−1

0

1

2

Hei

ght

/9.

51×

108

cm

ws0446

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0αml

Figure 7. Convective mixing length parameter αml computed lo-

cally from horizontally-averaged simulation data for upper branch

simulation ws0446 from Hirose et al. (2014), as a function ofheight and time. The αml data (already computed from quan-

tities smoothed over 1 orbit) has been smoothed by an additional

0.2 orbits in time to improve clarity. White regions within twopressure scale heights are either convectively stable, or are re-

gions for which our Newton-Raphson method to solve equations

(14)-(18) failed to converge within 100 iterations. Data outsidetwo pressure scale heights are discarded. The white gaps approx-

imately centered on the 55 orbit, 110 orbit, and 140 orbit epochsare times when radiative diffusion dominated convection in the

vertical transport of heat.

Note that β and γ0 are implicit functions of αml. Fconv, CP ,ρ, T , gz = −Ω2z, HP , ∇, ∇ad, and κ are read or computedfrom the MHD simulations. These variables are then hori-zontally averaged and smoothed by 1 orbit in time beforeαml is computed.

The results are shown in Figure 7. The epochs that arewhite at all heights in this figure are epochs when radiativediffusion dominates, and there is little vertical convectivetransport of heat. Most of the epochs that are actually con-vective have very reasonable values of αml that are of orderunity. It is generally only near the epochs that are radiativethat αml takes on substantially larger values.

The vertical profiles of the simulation data shown inFigure 6 average over both the convective and radiativeepochs, so it is not surprising, given the behavior shownin Figure 7, that unusually large mixing length parametersare required to describe these profiles with a pure convectivetransport treatment.

From the discussion here it is clear that time averagingthe intermittent convection obscures some of the physics dis-covered in the MRI simulations of Hirose et al. (2014). It isunclear how much this simplification affects the outcome.The duration of the convective limit cycle is ∼ 50 orbits,so for timescales & 1 day, this averaging procedure shouldbe a reasonable approximation. It is during the rapid tran-sition from quiescence to outburst that this simplificationbecomes questionable, making the inability of the DIM tocapture this time dependent behavior a clear limitation, butnothing better can be done in the framework of the standardDIM.

c© 2015 RAS, MNRAS 000, 1–18


3.4 Summary of MRI Simulation-Based VerticalStructure Equations and BoundaryConditions

The DIM framework uses the thermal pressure to provideboth the vertical hydrostatic support and to specify the ver-tical dissipation profile. In the simulations, magnetic pres-sure support can dominate thermal pressure near the pho-tosphere, although the magnetic to thermal pressure ratio isat most 1.5α . 20% near the midplane. Including magneticpressure in a single pressure framework is complicated, asit requires a modification of the alpha relation (equation 8)and it complicates the temperature gradient (equation 12)used in mixing length theory. We have therefore neglectedthis additional aspect of the simulation physics. Addition-ally, the DIM uses different, albeit similar, equation of stateand opacity tables.

With our modifications to the DIM our vertical struc-ture equations become

dP

d z= −ρΩ2z (19)

d ς

d z= 2ρ (20)

d lnT

d lnP= ∇ (21)

dFzd z

= Q+ = A0ΩP0

(P

P0

)δ(22)

A0 =3

2αP δ−1

0

∫∞0P dz∫∞

0P δ dz

, (23)

where ς(z) is the surface mass density between ±z, and ∇is determined by equation (12). Equations (22) and (23)are equivalent to our dissipation fit and the alpha relation(equations 7 and 8, respectively). Our midplane boundaryconditions are

z = 0 (24)

Fz = 0 (25)

ς = 0 (26)

T = T0 (27)

P = P0, (28)

and our exterior boundary conditions are

κRP =2

3Ω2z (29)

Fz = σT 4 (30)

ς = Σ, (31)

where κR is the Rosseland mean opacity.

3.5 Thermal equilibria: the S-curves

Before presenting outburst light curves based on the physi-cal models discussed in the previous two sections we brieflydiscuss the properties of the disc’s thermal equilibria. Weconsider two MRI-based models, DIMa and DIMRI, andcompare them to the standard DIM. DIMa adopts the samevertical structure assumptions as the standard DIM, butuses the MRI-based α(Teff) prescription of equations (5)-(6)discussed in section 2. DIMRI also uses this MRI-based α

102 103

Σ(g/cm2)

103

104

Teff

(K)

Σ+crit, T

+eff,crit

Σ−crit, T−eff,crit

DIM

DIMa

DIMRI

MRI Sims

5 10 15

100× α

Figure 8. Left: Loci of thermal equilibria in the Teff vs.

surface mass density Σ plane (the “S-curve”) at radius R =1.25 × 1010 cm for the standard DIM (blue), DIMa (magenta)

and DIMRI (green). The MRI simulation results are gray crosses

(one point for each stable simulation). Additionally, the criticalpoints (Σ+

crit, T+eff,crit) and (Σ−crit, T

−eff,crit) are marked for DIM.

Right: The MRI-based α (Teff) fit from Figure 1 plotted sideways

in solid black (DIMa and DIMRI both use this fit) with the re-sults from the same MRI simulations from the left plotted as gray

crosses. Equation (1) for the standard DIM, with αc = 0.03 andαh = 0.12, is plotted as the dotted black line.

prescription, combined with the MRI-based vertical struc-ture equations (19)-(31) summarized in section 3.4. Figure 8illustrates the differences in the S-curves produced by thesemodels at radius R = 1.25× 1010 cm around a 0.6M whitedwarf.

The most important parameters emerging from theS-curves shown in Figure 8 are the critical surface massdensities (Σ+

crit, Σ−crit) and effective temperatures (T+eff,crit,

T−eff,crit) at the ends of the (upper, lower) branches, mark-ing the points where (cooling, heating) transitions occur.In particular, the quotient of critical surface mass densities,QΣcrit ≡ Σ−crit/Σ

+crit, plays an important role in determining

the shape of the outburst lightcurve. It is therefore worthnoting that QΣcrit is different for all of the S-curves, withDIM having the largest QΣcrit. Despite our efforts, there isstill a basic discrepancy between the actual MRI simulationdata and the DIMRI S-curve in Figure 8. Both branches ofthe simulation S-curve extend a little further in Σ, leadingto a larger QΣcrit compared to that of DIMRI, and T−eff,crit issignificantly larger in DIMRI on the lower branch. The mis-match on the upper branch can be explained by our choiceof αml = 6. As an annulus in outburst approaches the criti-cal point (Σ+

crit, T+eff,crit) the role of convection increases and

presumably αml = 6 becomes increasingly less adequate.This is because the high value of αml that we adopt is dueto radiation dominated episodes in the intermittent convec-tion, which become less prominent as the annulus reachesthe end of the upper branch. One possible solution wouldbe to decrease αml towards the tip of the upper branch inthe DIMRI, and this may be worth exploring in the future.We show how adopting a smaller constant value of αml inDIMRI affects the outburst lightcurves below.

There are several issues which contribute to the mis-match between the simulation data and DIMRI on the lower

c© 2015 RAS, MNRAS 000, 1–18


branch, and these all stem from the fact that we have beenunable to find stable thermal equilibria in the simulationsfor effective temperatures higher than 3000 K. At such lowtemperatures, the opacities are so small that these equilib-ria are only marginally optically thick, with midplane opticaldepths τ . 5. This may be a problem given that the sim-ulations assume flux limited diffusion which may not accu-rately treat radiation transport at such low optical depths.The corresponding DIM models (i.e. DIM, DIMa, DIMRI)at these low temperatures also have low optical depths. Asa consequence, the density at the photosphere (the exteriorboundary condition) is a significant fraction of the centraldensity implying that a large fraction of mass (∼ 10− 50%)is ignored/neglected7. This missing mass likely has a signifi-cant impact on the temperature and density profiles of DIMmodels and is, at least partly, responsible for the miss-matchbetween the DIMRI curve and the simulation data on thelower branch of the S-curve. The influence that this miss-ing mass has on the temperature profiles may also explainwhy the DIMRI models near the end of the lower branchare convective (which is why the DIMRI S-curve turns upat Σ ≈ 180 g cm−2 in Figure 8), while there is no convec-tion in any of the lower branch MRI simulations. Again, wehave been unable to find any stable thermal equilibria athigher temperatures on the lower branch, where the opaci-ties would be higher and missing mass would be less of anissue in the DIM models. However, there is perhaps an evenbigger problem with the simulations on the lower branch,and that is that they neglect non-ideal MHD effects. As wedemonstrate in Appendix A, Ohmic resistivity and the Hallterm are likely to be important here, and so this also castsuncertainty on the lower branch results, and specifically thecritical point of the lower branch (Σ−crit, T

−crit) which con-

tributes to the determination of QΣcrit. Future work willhave to account for these effects on the lower branch. Ourfocus here, however, is to see whether the variation in α onthe upper branch that has been found in the simulations canproduce reasonable dwarf nova light curves.

Coming to the differences between our variants of theDIM, we first note that Σ−crit and T−eff,crit at the end of thelower branch are much higher on the classic DIM S-curvecompared to both the DIMa and DIMRI S-curves, conse-quently DIM has the largest QΣcrit. As DIMa shares ex-actly the same vertical structure assumptions and equationsas the classic DIM, this can only be due to the differentalpha-prescriptions between the two models (equations 5-6 and equation 1, respectively). Recall from Figure 1 thatour fit to the simulation data has α starting to increaseat Teff ∼ 4000 K, roughly indicating the end of the lowerbranch. By contrast, the choice of T+

crit in the classic DIMequation (1) corresponds to a much higher effective temper-ature (∼ 6000 K) at the end of the lower branch.

It should be noted that we have no simulation data for3000 K. Teff .7000 K in Figure 1, precisely because the sim-ulations failed to produce stable thermal equilibria in this

7 These inaccuracies do not affect standard DIM models of realoutbursts, because in these models the disc never cools down totemperatures at which these discrepancies appear. In the stan-

dard DIM one simply tunes αh and αc to obtain the requiredratio of the critical surface densities.

range. Therefore, our fit in equations 5-6 has some flexibilityin this temperature range. In principle, we could have fit thesimulation data in Figure 1 with a function that keeps α lowuntil the effective temperature increases above 6000 K, andthat would bring the lower branches of the DIMa and DIMS-curves into much better agreement. However, simulationws0467 shown in Figure 4, was started near the end of thelower branch at Teff ' 3000 K and underwent runaway heat-ing8. Hence our fits (equations 5-6) produce S-curves thatbetter represent the behavior observed in the simulations.

Another alternative to achieve agreement on the lowerbranch is to reduce Tc0 in the classic DIM. This has ac-tually already been done by Hameury (2002), who modi-fied this parameter to 8000 K, thereby producing a lowerbranch that only extended up to an effective temperatureof 3000 K. This modification produces very similar outburstlight curves, except that the quiescent light curves are flatterin shape, which actually may agree somewhat better withobservations (Hameury 2002). Whether the low T−eff,crit canbe claimed as a success of the MRI simulations will require afull treatment of the non-ideal MHD effects that have so farbeen neglected, but are likely to be crucial in the quiescentstate, and hence will likely shift the end of the lower branch.

As we discuss in more detail below, the new simulation-based vertical structure equations (most importantly the dif-ferent αml) are the reason DIMRI has a different location forthe end of the upper branch than DIM and DIMa in Fig-ure 8. In fact, if we plot a DIMRI S-curve with αml = 1.5,the end of the upper branch matches that of the DIM andDIMa S-curves. The very large effective mixing length pa-rameter αml = 6 used in DIMRI results in more efficientconvective transport, which flattens the temperature profilethus increasing Teff for a fixed midplane temperature Tc. Be-cause the opacity is largely determined by Tc, it is Tc whichdetermines where the end of the upper branch occurs. Forradiative cooling the effective and central temperatures arerelated through

Tc =

(3τtot

8

)1/4

Teff , (32)

where τtot is the total (vertical) disc opacity (see e.g. Dubuset al. 1999; Kotko et al. 2012). For a fully ionized disc theRosseland opacity is κ ∼ ΣH−1Tc

−7/2cm2/g from whichfollows the well known (Shakura & Sunyaev 1973) rela-tion Teff ∝ Σ5/14. Recombination breaks this relation andchanges its slope while convection enhances cooling. Theresult is that the upper branch ends at higher effective tem-perature, and hence higher surface density.

4 OUTBURST LIGHT CURVES

4.1 The outburst cycle

During the low luminosity “quiescent” phase of a dwarf novacycle, the effective temperature in the whole disc is lower

8 Although this helps constrain the end of the lower branch, it is

numerically challenging to further resolve the critical point (Σ−crit,

T−crit), as the simple act of relaxing from the initial conditions

could push a simulation too far from (Σ−crit, T−crit) to obtain a

thermal equilibrium.

c© 2015 RAS, MNRAS 000, 1–18


than T−eff,crit (the disc is cold). The mass accretion rate isnot constant across the cold disc so the disc accumulatesmatter, increasing its temperature and surface density (eachdisc annulus moves up the lower branch of its local S-curve).Finally, at some radius, R0, the accumulation time becomesshorter than the viscous time, and the ionization of gas be-comes so significant that the local cooling mechanisms be-come inefficient and thermal equilibrium is lost. The annulusat R0 undergoes rapid heating and makes the transition tothe hot state (upper branch of its local S-curve). A narrowfully-ionized region of high viscosity has just formed at R0,and is surrounded by cold matter. This induces a steep ra-dial temperature gradient and the formation of the heatingfront, indicating the beginning of an outburst. Additionally,a spike in the Σ profile arises as a consequence of differentviscous efficiencies in the hot and cold parts of the disc: thelow viscosity outside the hot annulus provides insufficientoutwards angular momentum transport to prevent furtheraccumulation of mass at R0. These steep Σ and Teff gradi-ents in the heating front cause matter and heat to diffuseto the adjacent annuli, forcing their transition to the hotstate. The hot region in the disc widens as the heating frontpropagates through the disc causing the luminosity to rise.The elevated mass accretion rate in the hot region reducesthe surface density behind the heating front and enhancesthe mass inflow to the inner disc region9.

Once the heating front reaches the outer disc edge, thedisc is fully ionized and reaches its maximum brightness(the outburst maximum). In the DIM, the minimum crit-ical surface density Σ+

crit at the end of the upper branchof the local S-curve is approximately proportional to ra-dius R (Hameury et al. 1998), causing Σ+

crit to be highestat Rout. Consequently Σ manages to only rise slightly abovethis critical value near Rout as the heating front passes andit falls below the critical value almost immediately after thefront dissipates. At the radius where this happens the cool-ing is strongly enhanced by the change of opacities whenTeff <T+

eff,crit and Σ and Teff gradients lead to the formationof the cooling front. The inward propagation of the coolingfront through the disc leads to the observed outburst decay.

The cooling front develops at the outer disc edge almostat the same time as the heating front disappears, allowing notime for the mass accumulated in the outer parts of the discto arrive at the inner disc radius before the cooling front setsin, even though the mass accretion rate everywhere in thehot disc has increased beyond the mass transfer rate fromthe secondary. The surface density profile at the outburstmaximum is not yet proportional to R−3/4 as expected ina hot stable disc. Therefore, even after the development ofthe cooling front, the mass accretion rate near the inner discedge still increases until the mass excess from the outer discregion has traveled through the whole disc and has beenaccreted onto the white dwarf (see Figure 13 and for moredetails Kotko & Lasota 2012). Only after this will the hotregion ahead of the cooling front reach the hot stable state

9 We describe here an inside-out outburst. For high mass-transferrates, outbursts can be of outside-in type, i.e. the heating frontpropagates inwards from the disc outer regions (see e.g. Lasota

2001).

where the constant mass accretion rate is of order Mtr, themass transfer rate from the secondary.

4.2 Reflares

Due to the high Mtr and low αc, the disc may accumulatea lot of mass during the quiescent phase and rise to an out-burst. If the viscosity in the hot disc is not efficient enough(i.e. when αh is relatively low) to redistribute the mass ex-cess accumulated during the previous outburst phases tothe inner region (where it can be accreted), or if the criticalpoints (Σ−crit, T

−crit) and (Σ+

crit, T+crit) are too close (i.e. QΣcrit

is too small), the fronts propagating in the disc (both cool-ing and heating) may be stopped before arriving at eitherof the two disc edges. This gives rise to the appearance ofreflares in the outburst lightcurves. As we discuss below,all our MRI-based models exhibit this phenomenon, and wetherefore begin with a brief description of the cause: it is theconfluence of the small αh high on the upper branch as wellas the small QΣcrit that is responsible for these reflares.

As a cooling front propagates inward, the high viscosityof the hot matter inside the cooling front contrasted with thelow viscosity outside the cooling front causes an outward dif-fusion of matter across the front. This in turn causes a deficitin surface density within the front itself, followed by an en-hanced surface density in the cold region behind (outside)the front. Hence, as the cooling front moves inward in radius,at some point the post-front Σ may become high enough tocross the critical value Σ−crit at the end of the lower branchof the local S-curve. It is here where it is clearest that thelow αh and the lack of sufficient separation between Σ−crit

and Σ+crit (i.e. too small QΣcrit) conspire against the smooth

propagation of a cooling front by creating a mass excessoutside the front and setting a low critical threshold respec-tively.

In this situation a new heating front arises and startsto move outwards. The matter heated by this newly formedfront flows at a high rate into the zone of the cooling front,increasing its temperature and surface density. This inflowof hot gas eventually destroys the cooling front and onlya heating front is left. As a result, the inward propagat-ing cooling front behaves as if it is reflected into an outwardpropagating heating front before it arrives at the inner edge.A similar mechanism can cause reflection of the heating frontpropagating toward the outer disc radius. If the post-front Σremains close to Σ+

crit the elevated accretion rate in the hotregion behind the front may cause Σ to fall below the criti-cal value and a cooling front will start to form. The reducedtransport of the angular momentum through the emergingcold zone finally stops the propagation of the heating frontand a newly formed cooling front will move inward. These re-flections produce a reflare pattern in the outburst lightcurves(see sawtooth-like features in Figure 9), which are not ob-served in standard dwarf novae. As reviewed in section 4.3of Lasota (2001), reflares are a common feature of the DIM,and one must work to get rid of them by choosing appropri-ate values of αh and αc in order to agree with the smoothobserved light curves. As we will see shortly, reflares are alsoa generic feature of all our MRI-based lightcurves. However,it is important to reemphasize that the reflares (and the de-tails of outbursts) are dependent on where the lower branchends; a detail we do not claim to model accurately.

c© 2015 RAS, MNRAS 000, 1–18


140 160 180 200 220

Time (days)

9

10

11

12

13

14

15

16

17

MV

a) DIM-5e14, Mtr = 5.0×1014

160 180 200 220 240

Time (days)

7

8

9

10

11

12

13

14

MV

b) DIM-1.5e16, Mtr = 1.5×1016

940 960 980 1000 1020

Time (days)

9

10

11

12

13

14

15

16

17

18

MV

c) DIMa-5e14, Mtr = 5.0×1014

1520 1540 1560 1580 1600 1620 1640 1660

Time (days)

8

9

10

11

12

13

14

15

MV

d) DIMa-1.5e16, Mtr = 1.5×1016

60 80 100 120 140 160

Time (days)

8.0

8.5

9.0

9.5

10.0

10.5

11.0

11.5

MV

e) DIMRI-5e14, Mtr = 5.0×1014

100 105 110 115 120 125

Time (days)

11.5

12.0

12.5

13.0

13.5

14.0

MV

f) DIMRI-1.5e16, Mtr = 1.5×1016

Figure 9. Visual magnitude lightcurves for six of the models listed in Table 1. The sawtooth-like features are reflares. The units for the

listed mass transfer rates are g/s.

c© 2015 RAS, MNRAS 000, 1–18


Model Mtr [g/s] Aoutb Toutb Tquiesc Figure

DIM-5e14 5× 1014 5.4 2.3 47.3 9a

DIM-1.5e16 1.5× 1016 4.6 6.3 11.6 9b

DIMa-5e14 5× 1014 7 11.6 46.4 9c

DIMa-1.5e16 1.5× 1016 6.3 50 0 9d

DIMRI-5e14 5× 1014 1.7 2.8 0 9e

DIMRI-1.5e16 1.5× 1016 2.6 19.9 0 9f

DIMRI-αML = 1.5 5× 1014 7.7 17.3 60.1 10

DIMRI-6e13 6× 1013 3.2 1.9 7 11

DIMRItr-1.5e16 1.5× 1016 3.4 14.6 4.7 12

Table 1. The parameters of the outbursts measured in our calculated lightcurves. Aoutb is the outburst amplitude in magnitudes, Toutb

is the outburst duration time in days and Tquiesc is the quiescence time in days, i.e. the time elapsed between the end of an outburstand the beginning of the next. The last column lists the figure where the lightcurve for a given model can be found.

4.3 Results

The outburst properties depend on disc viscosity and theparameters characterizing a binary. For the new DIMRI tobe considered as a possible replacement for the standardDIM, it should reproduce the basic features of dwarf novaelightcurves such as outburst amplitude, outburst durationand quiescence duration.

To better understand how the new α-prescription andthe new disc vertical structures introduced into the classi-cal DIM influence outburst light curves, we calculated theselight curves using three models: DIMRI, DIM with α as afunction of Teff (DIMa) and classic DIM with αc = 0.03and αh = 0.12. All models were calculated for the sameset of parameters: a primary mass M1 = 0.6 M, the discinner radius Rin = 8.67 × 108 cm, and the circularizationradius Rcirc = 2.85 × 1010 cm. In addition we run the cal-culation for two different values of the mass transfer rate:Mtr = 1.5× 1016 g/s and Mtr = 5× 1014 g/s for each model(see Table 1). In all models the outer disc radius is vari-able due to the fact that we take into account the tidalforce acting between the secondary star and the disc. Forthe models in this paper, the average outer disc radius is〈Rout〉 = 4.6× 1010 cm.

Analyzing the differences between the lightcurves (seeFigures 9-12) gives insight into the physical implications ofour modifications to the DIM. One subtle difference betweenthe classic DIM lightcurves and the MRI based lightcurvesis that the MRI based ones are not strictly periodic. Wedo not understand why the new α prescription causes this.One possibility is that the disc needs much more time torelax with this prescription. Conversely, the most strikingdifference between the DIM and the two other model lightcurves is that the outburst decay in DIM is smooth while inDIMRI and DIMa the outburst decay has small amplitudebrightness variations characteristic of reflares (compare Fig-ure 9c-f with Figure 9a,b). The reason that reflares do notappear in the DIM but are present in the two other mod-els is connected to our α-prescription, but is also tied toour uncertainties on the lower branch, specifically the loca-tion of the critical point (Σ−crit, T

−crit). In the DIM, QΣcrit is

larger than the other models and α maintains its high value(αh = 0.12 αc) in the whole hot part of the disc until thecooling front passage. In the models where α is a functionof effective temperature, the higher viscosity is present in amuch more narrow region, i.e. in which the central tempera-ture is higher than T−crit but lower than 5×104K. Therefore,the mass in the inner disc region is accreted at a lower ratethan in the DIM due to the lowered viscosity in the hightemperature regime, leading to an excess of mass behind(outside) cooling fronts. It is the combination this effect cou-pled with the small QΣcrit which is directly responsible forthe reflares seen in DIMa and DIMRI. Furthermore, QΣcrit

as determined by MRI simulations is actually larger thanthat found in DIMRI, which suggests that DIMRI is moresusceptible to reflares than what the simulation data imply.

Figure 13 shows how the higher surface mass density inthe inner disc and smaller QΣcrit in DIMa leads to reflec-tions of the inward propagating cooling front and reflares.The first epoch shown (the red curves labeled 1) correspondsto the time when the outward propagating heating front hasjust arrived in the outer disc, which is why there is a spikein surface density and bump in midplane temperature atR ∼ 1.3 × 1010 cm. By this time an inward propagatingcooling front has already been launched, and is located at' 7.6 × 109 cm where the surface density has reached thecritical surface density Σ+

crit on the upper branch of the localS-curve. As the front propagates inward from 1 (red) to 2(green) to 3 (blue), the gradients in viscosity cause outwardmass diffusion, thereby producing a rarefaction in surfacedensity down to Σ+

crit within the cooling front, followed byan enhanced surface density behind (outside) the front. Be-cause the inner disc in the outburst state has such a highsurface density due to the low values of α high up on theupper branch relative to the DIM, the post-front excess insurface mass density is also high, and eventually, at epoch 4(magenta) at R = 4.6× 109 cm, reaches the critical surfacedensity Σ−crit at the end of the lower branch of the local S-curve (highlighting the role of small QΣcrit). This triggers aheating front which then propagates outward, as evident inepoch 5 (black) at R = 5.3× 109 cm.

Hence, directly as a consequence of the small Σ−crit (or

c© 2015 RAS, MNRAS 000, 1–18


500 550 600 650 700

Time (days)

9

10

11

12

13

14

15

16

17

18

MV

DIMRI, αML = 1.5, Mtr = 5.0×1014

Figure 10. Light curves calculated from DIMRI but for αml

= 1.5; Mtr = 5.0× 1014 g s−1. Compare with Fig. 9e.

30 40 50 60 70

Time (days)

13.0

13.5

14.0

14.5

15.0

15.5

16.0

16.5

17.0

MV

DIMRI-6e13, Mtr = 6.0×1013

Figure 11. Light curves calculated from DIMRI with Mtr =6.0× 1013 g s−1.

alternatively small QΣcrit) combined with the low α’s andresulting higher surface densities in the inner disc in out-burst, the cooling front that would normally cause a tran-sition back to quiescence propagates instead with difficulty,through a sequence of reflections seen as reflares in the lightcurve. In contrast, in the DIM with suitably chosen αh andαc, there is a larger Σ−crit making it harder to trigger a re-flare. Additionally, the mass diffusion and accretion duringthe outburst is much higher and the inner disc is able to pro-cess sufficient mass to lower the inner surface density andavoid the appearance of the reflares during the outburst de-cay. The contested propagation of the cooling front in DIMacauses the outburst decay phase to last longer compared toDIM (for example compare Figure 9a with Figure 9c). Dur-ing this time more mass is being accreted onto the whitedwarf in the DIMa, leaving the disc less massive and lessluminous than in the DIM at the end of outburst. This re-sults in a higher amplitude outburst for the DIMa, whichhighlights the effect reflares have on outbursts.

500 520 540 560 580 600

Time (days)

8

9

10

11

12

13

14

MV

DIMRItr-1.5e16, Mtr = 1.5×1016

Figure 12. Light curves calculated from DIMRI with inner disc

radius truncated by the magnetic field with magnetic momentµ = 8× 1030G cm3 for αml = 6.0; Mtr = 1.5× 1016 g s−1. As can

be seen by comparing this to Fig. 9f, the truncation of the innerdisc results in the appearance of quiescence and more regular

outbursts.

The difference in light curves between DIMRI and DIMais due to the different αml values and different dissipationprofiles. The mixing length parameter αml is the most impor-tant difference, as can be seen by comparing Figure 9e withFigure 10, which presents the same DIMRI calculation butwith αml restored to its traditional value of 1.5. Convectionsets in close to the point where the cold branch of the S-curveends which leaves this point sensitive to αml. This effect iseven stronger on the critical point where the hot branchstarts, as this is where convection is the strongest. Highervalues of αml shift both critical points at which the S-curvebends closer together, leading to a smallerQΣcrit. This altersthe global behavior of the disc and the outburst properties:the decay from the outburst in a disc with αml = 6 starts athigher Σ and higher Teff , and less mass is accreted and ac-cumulated in the disc during the outburst cycle. Therefore,more efficient convection produces outbursts that are morefrequent and of lower amplitude, and even lack quiescentphases. While it is important to note that this lack of quies-cence may be related to uncertainties in the end of the lowerbranch, there are other ways out. In the discussion below,we examine a few options to restore or modify quiescence.

The comparison of DIMRI with αml = 1.5 and DIMa(Figure 10 and Figure 9c) highlights the importance ofchanging the dissipation profile, as this is the only differ-ence between these two models. The main difference is thatthe outbursts in DIMRI with αml = 1.5 are wider and thequiescence is longer than in the DIMa light curve.

Changing the mixing length parameter is not the onlyway to increase the outburst amplitude and restore quiescentphases to the DIMRI models. Increasing the mass transferrate, while all other parameters are fixed, makes the dischotter and denser. This means that the surface density ev-erywhere in the quiescent disc is closer to the critical valueand the disc luminosity is higher. The result is more frequentlower amplitude outbursts for higher mass transfer rate re-gardless of model, as illustrated by comparing the two mass

c© 2015 RAS, MNRAS 000, 1–18


101

102

Σ(g

/cm

2)

Σ−critΣ+

crit

103

104

105

Tc

(K)

109 1010

R (cm)

0.05

0.10

α

934 936 938 940 942 944 946 948

Time (days)

8

10

12

14

16

18

20

MV

12 3

4

5

Figure 13. Left panels: Radial profiles of surface density (top), midplane temperature (middle), and α-parameter (bottom) during theinitial decay from outburst for the DIMa calculation with M = 5× 1014 g s−1. Right panel: Zoom of the first outburst in the lightcurve

Figure 9(c). Different successive times 1-5 are shown by different colors as indicated, with 1 (red) being the first and 5 (black) being the

last. It is important to note that α is small for R < 3× 109 cm for all epochs shown despite the fact that the disc is in outburst, becausewe are high on the upper branch here. This results in the elevated surface mass density that can be seen in the inner disc. As the cooling

front (the dip/rarefaction in Σ located at R ≈ 7.6× 109 cm in epoch 1) propagates inwards through the disc, mass is redistributed from

ahead of the front where Σ is high to the post-front region. Eventually, the post-rarefaction surface density crosses the critical surfacedensity Σ−crit at the end of the lower branch of the local S-curve. This occurs approximately at epoch 4 (magenta), which is when the

inward propagating cooling front is reflected into the heating front seen at R ≈ 5.3×109 cm in epoch 5 (black). This figure clearly shows

that either further separating Σ−crit and Σ+crit (i.e. increasing QΣcrit) or reducing the excess mass in the inner disc by increasing α high

on the upper branch could help alleviate reflares.

transfer rates shown in Figure 9 for each of our three models.Hence merely reducing the mass transfer rate in the DIMRImodel can increase the outburst amplitude and restore qui-escent phases. If one sets Mtr as low as Mtr = 6× 1013 g/sin the DIMRI (with αml = 6) the elapsed time between twoconsecutive outbursts starts to be longer and also the out-burst amplitude rises (see Table 1 and Figure 11). Note thatfor such a low Mtr, the discs in DIM and DIMa become coldand stable.

There is yet another alternative way to restore quies-cence in the DIMRI light curves. Observed X-ray fluxes inquiescent dwarf novae are far too large compared to whatmodels predict. A solution to this problem is a truncationof the inner disc. Numerical calculations by Hameury et al.(2000) and Schreiber et al. (2003) confirm that truncatingthe inner disc has substantial influence on the dwarf novaelightcurves and may solve the discrepancy between obser-vations and theory. This truncation may be caused by themagnetic field of the primary white dwarf. A white dwarfmagnetic field in the range 104 − 107G (which translates toa magnetic moment µ ≈ 1− 103 × 1030 G cm3) is sufficientfor the magnetic pressure close to the white dwarf to exceedthe gas and ram pressures of the infalling matter duringthe quiescent phase of the dwarf nova cycle. The inner discradius is therefore pushed away from the white dwarf to a

radius RM (e.g. Frank et al. 2002):

Rin = RM = 9.8× 108Mtr−2/715 M

−1/71 µ

4/730 cm (33)

where µ30 is the magnetic moment in units of 1030 G cm3,M1 is the mass of the primary in solar masses and M15 is themass accretion rate in units of 1015 g/s. During outburst thesituation changes, as the higher mass accretion rate sharplyincreases the ram pressure of matter which then dominatesthe magnetic pressure, and the inner edge of the disc ap-proaches the surface of the white dwarf. Taking into ac-count the variation in inner disc radius according to Eq.(33)restores the quiescent phase in the simulated DIMRI lightcurves (compare Figures 9f and 12).

5 DISCUSSION

There are four main aspects of observed dwarf nova lightcurves which need to be reproduced in order to have a suc-cessful theoretical model: quiescence duration, outburst am-plitude, outburst duration, and shape. For the most part ourMRI based models DIMa (which differ from the DIM mod-els only by incorporating the MRI simulation based α (Teff))and DIMRI can reproduce these attributes, with the notableexception of reflares. These reflares are the result of two con-tributing factors. Namely, the small ratio of the surface den-sities at the ends of the lower and upper branches, QΣcrit,

c© 2015 RAS, MNRAS 000, 1–18


and the low value of α found high up on the upper branch.This low α causes an excess of surface mass density Σ inthe inner disc. Consequently, as a cooling front propagatesinward through the disc it accumulates substantial mass inthe post-front region, and due to the small QΣcrit, this post-front excess in Σ easily surpasses the critical value, Σ−crit,and initiates a reflare. The reflares created by this series ofevents tend to prolong the decay from outburst. We havefound two mechanisms which help to hasten this elongateddecay from outburst: reducing the mass transfer rate andtruncation of the inner disc by a white dwarf magnetic field(Figures 11-12). However, these tweaks are merely attemptsto treat the symptoms caused by the greater underlying is-sues mentioned above10.

It is important to remember that QΣcrit is determinedby the physics of both the upper and lower branches, andthe lower branch physics is very uncertain. The simulationswere produced with the ideal MHD and flux limited dif-fusion approximations. While these are reasonable on theupper branch, the lower branch is a different story. Opticaldepths along the lower branch are low (τ . 5) bringing intoquestion our usage of the flux limited diffusion approxima-tion. Moreover, non-ideal effects, particularly resistivity andHall effects (Bai 2014; Lesur et al. 2014, Appendix A) areimportant on the lower branch where the ionization fractionis low. This strongly motivates the need to pursue non-idealMHD simulations of the lower branch, and as in protostel-lar discs (Igea & Glassgold 1999), it may be necessary toaccount for irradiation by ionizing x-rays from the bound-ary layer between the white dwarf and the accretion disc,because any source of ionization has the potential to signifi-cantly modify non-ideal MHD effects. MRI turbulence mayonly exist in the irradiated surface layers, possibly leavinga magnetically driven laminar flow in the resistive “deadzone” interior (Turner & Sano 2008). Small amounts of hy-drodynamic angular momentum transport associated withthermal convection (Lesur & Ogilvie 2010) may also exist.However, it may turn out that none of these local mecha-nisms is sufficient to explain the quiescent state, and thatsomething involving more global physics is required. For ex-ample, recent isothermal and adiabatic global simulationsby Ju et al. (2016) suggest that spiral waves excited by thetidal field of the donor star may contribute significantly tothe angular momentum transport in the quiescent state, andthis physics cannot be captured by a local stress-pressure re-lation. All this speculation, again highlights the uncertaintyin QΣcrit, and the need for the inclusion of non-ideal effectson the lower branch.

Even with this uncertainty, however, we have found thatenhancement of α only near the end of the upper branch,where thermal convection happens, is capable of produc-ing outbursts. This outcome is non-trivial and suggests thatconvection plays a major role in the outbursts of dwarf no-vae, and while important, convection may not be the end ofthe story. The low values of α high up on the upper branchas found by Hirose et al. (2014), are a contributing factorin the appearance of reflares. As mentioned early on in thispaper, MRI simulations with net vertical magnetic flux have

10 Notice, however, that disc truncations is independently re-

quired by dwarf nova observations(see e.g., Lasota 2001).

larger α values (Hawley et al. 1995; Sano et al. 2004; Pessahet al. 2007). While we have considered a way of increasing αwith zero net vertical magnetic flux by the effects of hydro-dynamic convection on MRI turbulence, there is no reasonto view these explanations as mutually exclusive. Accord-ingly, simulations examining how net vertical magnetic fluxmodifies convection and the associated enhancement of αfound in Hirose et al. (2014) would be useful and may helpalleviate reflares.

It is also important to note that we have not success-fully incorporated all aspects of the physics observed in thesimulations in our attempts to model outburst light curveshere. The MRI-based physics that we have succeeded in in-corporating into the DIM has not produced S-curves thatcompletely agree with the simulation data (cf. Figure 8).Both the upper and lower branches of the MRI simulation S-curve are extended slightly further in Σ compared to DIMRI.This implies that DIMRI should have a larger QΣcrit whichin turn would increase the amplitude and the quiescenceduration and may reduce or even alleviate the reflares.

Moreover, it is clear that the MRI simulations exhibitsome time dependent behavior (e.g. intermittent convection)that the DIM simply is incapable of handling. During theconvective epochs the Shakura & Sunyaev (1973) α param-eter is enhanced and also αml ∼ 1.5. To re-emphasize this, αis significantly higher and αml significantly lower during theintermittent convective epochs than their respective timeaveraged quantities. This has a significant impact on theevolution of an annulus as it transitions from quiescence tooutburst11. Thermally unstable MRI simulations that areheating towards the hot branch are fully convective, withno intermittency. Therefore, when an annulus first makes itto the hot branch it will be fully convective and “see” an in-stantaneous S-curve which is characterized by the α and αml

seen during convective epochs. This annulus will then evolveand after a few thermal times become better characterizedby the time averaged values for α and αml and the asso-ciated “long-term” S-curve. Global MRI simulations of thepropagation of heating and cooling fronts would obviouslybe of great interest, but are not yet viable for the dwarf novaproblem.

We finally note that the quiescent state is also a prob-lem for the DIM models (i.e. DIM, DIMa, DIMRI) presentedhere and historically in the standard DIM; see e.g. Lasota(2001). Our lower branch DIM models with Teff < 3000 Khave low optical depths12 (τ . 5) which leads to inaccu-racies, most notably ∼ 10 − 50% of mass lying outside ofthe vertical boundary condition. However, until non-idealeffects are included in the lower branch simulations, it is fu-tile to try and bring the simulation data and DIM modelsinto better agreement here. This impacts the details of out-burst lightcurves, most notably the occurrence of reflares.While these reflares are not observed in the outbursts ofstandard dwarf novae, we nevertheless recover outburst time

11 We note that our shearing box simulations are local not onlyin the radial direction, but also in the azimuthal direction. There-

fore, the time dependent behavior may be less manifest in obser-

vations where azimuthal variations are averaged.12 This does not occur in the classical DIM, because without the

MRI-based constraints we have imposed in this paper αc and αh

can be tuned to avoid this problem.

c© 2015 RAS, MNRAS 000, 1–18


scales and amplitudes comparable to those observed, imply-ing that the simulation-based models of the upper branchare in good agreement with observations.

6 CONCLUSIONS

We have incorporated the convection-induced enhancementof α close to the hydrogen ionization temperature that wasdiscovered in MRI simulations by Hirose et al. (2014) intothe DIM. This, for the first time, places the early inferenceby Mineshige & Osaki (1983); Smak (1984); Meyer & Meyer-Hofmeister (1984) that α had to be larger in outburst indwarf novae on a strong and clear theoretical foundationbased on MRI turbulence. We have also shown how to incor-porate aspects of the time-averaged vertical structure (dissi-pation profiles, intermittent convective heat transport) intothe DIM. With suitable parameter choices and/or trunca-tion of the inner disc by a white dwarf magnetic field, theresulting lightcurves are able to produce outburst and qui-escent durations, as well as outburst amplitudes, that areconsistent with observations of dwarf novae. Further workto actually fit particular dwarf nova systems with this modelmight be worthwhile.

A generic feature of our MRI-based α-prescription is theappearance of reflares. These reflares are caused by the con-fluence of low α’s high up on the upper branch and the smallratio of critical surface mass densities QΣcrit. Our biggestuncertainty in this work is the physics of the lower branchand consequently QΣcrit. This uncertainty primarily stemsfrom the exclusion of important non-ideal MHD effects as-sociated with the largely electrically neutral plasma on thelower branch. This gives us a clear motivation and directionfor future work to address these inaccuracies in new simu-lations and to obtain a more realistic understanding of thelower branch and its end.

Additionally, while we have tried to incorporate MRI-based physics into the DIM, this has only been done forannuli in thermal equilibrium. We still do not understandhow MRI physics might affect the propagation of heatingand cooling fronts, and it is these that are responsible forreflares. MRI simulations in a more global geometry thatcan track the propagation of such fronts would of coursebe very illuminating, though such simulations appear to bechallenging with current resources.

Finally, it is important to remember that reflares arealso a common problem even in the standard DIM, whereone is free to choose values of αh and αc to try and getrid of them (e.g. Lasota 2001). This freedom is of coursean illusion, as the stresses are actually determined by theunderlying physics of turbulence in the disc. The fact thatapplications of our MRI simulation results to the DIM giverise to reflares helps sharpen this problem by relating it moreto the fundamental physics responsible for stresses in thedisc.

ACKNOWLEDGMENTS

We thank the anonymous referee for a constructive re-port that led to significant improvements in this paper. Wealso wish to acknowledge the seminal role of Jean-Marie

Hameury and Guillaume Dubus in developing the numericalmethods necessary to accurately model lightcurves withinthe DIM.

Additionally, we also thank Jean-Marie Hameury forconstructive feedback on this work. This research was sup-ported by the United States NSF grant AST-1412417, thePolish NCN grants UMO-2013/08/A/ST9/00795 and UMO-2015/19/B/ST9/01099, and by Japan JSPS KAKENHIGrants 24540244 and 26400224, joint research project ofILE, Osaka University. JPL was supported in part by a grantfrom the French Space Agency CNES. OB thanks the In-ternational Space Science Institute in Bern for its supportwhile this paper was being completed. Numerical calcula-tions were partly carried out on the Center for ScientificComputing from the CNSI, MRL: an NSF MRSEC (DMR-1121053) and NSF CNS-0960316, and the Cray XC30 atCfCA, National Astronomical Observatory of Japan.

REFERENCES

Bai X.-N., 2014, ApJ, 791, 137Bai X.-N., Stone J. M., 2013, ApJ, 767, 30Balbus S. A., Hawley J. F., 1991, ApJ, 376, 214Balbus S. A., Hawley J. F., 1998, Reviews of ModernPhysics, 70, 1

Balbus S. A., Terquem C., 2001, ApJ, 552, 235Blaes O., Krolik J. H., Hirose S., Shabaltas N., 2011, ApJ,733, 110

Blaes O. M., Balbus S. A., 1994, ApJ, 421, 163Davis S. W., Stone J. M., Pessah M. E., 2010, ApJ, 713,52

Draine B. T., 2011, Physics of the Interstellar and Inter-galactic Medium

Draine B. T., Roberge W. G., Dalgarno A., 1983, ApJ, 264,485

Dubus G., Hameury J.-M., Lasota J.-P., 2001, A&A, 373,251

Dubus G., Lasota J.-P., Hameury J.-M., Charles P., 1999,MNRAS, 303, 139

Frank J., King A., Raine D. J., 2002, Accretion Power inAstrophysics: Third Edition

Fromang S., Latter H., Lesur G., Ogilvie G. I., 2013, A&A,552, A71

Fromang S., Papaloizou J., Lesur G., Heinemann T., 2007,A&A, 476, 1123

Guan X., Gammie C. F., 2011, ApJ, 728, 130Hameury J., Lasota J., Warner B., 2000, A&A, 353, 244Hameury J.-M., 2002, in Gansicke B. T., Beuermann K.,Reinsch K., eds, The Physics of Cataclysmic Variablesand Related Objects Vol. 261 of Astronomical Society ofthe Pacific Conference Series, The disc instability model.p. 377

Hameury J.-M., Menou K., Dubus G., Lasota J.-P., HureJ.-M., 1998, MNRAS, 298, 1048

Hawley J. F., Balbus S. A., 1991, ApJ, 376, 223Hawley J. F., Gammie C. F., Balbus S. A., 1995, ApJ, 440,742

Hawley J. F., Gammie C. F., Balbus S. A., 1996, ApJ, 464,690

Hawley J. F., Guan X., Krolik J. H., 2011, ApJ, 738, 84Hirose S., 2015, MNRAS, 448, 3105

c© 2015 RAS, MNRAS 000, 1–18


Hirose S., Blaes O., Krolik J. H., Coleman M. S. B., SanoT., 2014, ApJ, 787, 1

Hirose S., Krolik J. H., Stone J. M., 2006, ApJ, 640, 901Igea J., Glassgold A. E., 1999, ApJ, 518, 848Ju W., Stone J. M., Zhu Z., 2016, ApJ, 823, 81King A. R., Pringle J. E., Livio M., 2007, MNRAS, 376,1740

King A. R., Ritter H., 1998, MNRAS, 293, L42Kotko I., Lasota J.-P., 2012, A&A, 545, A115Kotko I., Lasota J.-P., Dubus G., Hameury J.-M., 2012,A&A, 544, A13

Lasota J.-P., 2001, NewAR, 45, 449Lasota J.-P., Dubus G., Kruk K., 2008, A&A, 486, 523Lesur G., Ferreira J., Ogilvie G. I., 2013, A&A, 550, A61Lesur G., Kunz M. W., Fromang S., 2014, A&A, 566, A56Lesur G., Longaretti P.-Y., 2007, MNRAS, 378, 1471Lesur G., Ogilvie G. I., 2010, MNRAS, 404, L64Meyer F., Meyer-Hofmeister E., 1984, A&A, 132, 143Miller K. A., Stone J. M., 2000, ApJ, 534, 398Mineshige S., Osaki Y., 1983, PASJ, 35, 377Mineshige S., Wheeler J. C., 1989, ApJ, 343, 241Paczynski B., 1969, Acta Astron., 19, 1Pessah M. E., Chan C.-k., Psaltis D., 2007, ApJ, 668, L51Potter W. J., Balbus S. A., 2014, MNRAS, 441, 681Sano T., Inutsuka S.-i., Turner N. J., Stone J. M., 2004,ApJ, 605, 321

Sano T., Miyama S. M., Umebayashi T., Nakano T., 2000,ApJ, 543, 486

Schreiber M. R., Hameury J.-M., Lasota J.-P., 2003, A&A,410, 239

Shakura N. I., Sunyaev R. A., 1973, A&A, 24, 337Shi J., Krolik J. H., Hirose S., 2010, ApJ, 708, 1716Shi J.-M., Stone J. M., Huang C. X., 2016, MNRAS, 456,2273

Simon J. B., Beckwith K., Armitage P. J., 2012, MNRAS,422, 2685

Smak J., 1984, Acta Astron., 34, 161Smak J., 1999, Acta Astron., 49, 391Sorathia K. A., Reynolds C. S., Armitage P. J., 2010, ApJ,712, 1241

Sorathia K. A., Reynolds C. S., Stone J. M., Beckwith K.,2012, ApJ, 749, 189

Suzuki T. K., Inutsuka S.-i., 2009, ApJ, 691, L49Turner N. J., 2004, ApJ, 605, L45Turner N. J., Sano T., 2008, ApJ, 679, L131Turner N. J., Stone J. M., Krolik J. H., Sano T., 2003, ApJ,593, 992

van Paradijs J., 1996, ApJ, 464, L139Wardle M., 2007, Ap&SS, 311, 35

APPENDIX A: NON-IDEAL MHD EFFECTS INDWARF NOVA QUIESCENCE

Here we examine non-ideal MHD effects due to thermal ion-ization only (i.e. neglecting irradiation) and their possibleimportance in dwarf nova quiescence. By non-ideal MHDeffects, we are referring to Ohmic dissipation (i.e. resistiv-ity), the Hall term, and ambipolar diffusion. The full induc-tion equation including these terms can be written as (Lesur

et al. 2014)13

∂B

∂t−∇×(v×B)=− ∇

4π×(ηOJ + ηHJ×eb − ηAJ×eb×eb) ,

(A1)where J = 4π∇ × B and eb = B/ |B|. The terms on theright hand side are respectively the Ohmic resistive term(O), the Hall term (H), and the ambipolar diffusion term(A), where the η coefficients are the respective diffusiv-ity coefficients. The Hall term is caused by a velocity dif-ference between electrons (e) and heavy cations (i), whileOhmic dissipation and ambipolar diffusion are caused bycollisions between neutrals (n) and electrons or cations re-spectively. Thus, it is worthwhile briefly discussing the rel-evant quantities involving these collisions. The momentumrate coefficients for ion-neutral and electron-neutral colli-sions are < σv >ni= 1.9× 10−9 cm3 s−1 (Draine 2011) and< σv >ne= 8.3 × 10−10T 1/2 cm3 s−1 (Draine et al. 1983),respectively. The ion-neutral drag coefficient is therefore

γ =< σv >ni

mn +mi= 2.7×1013

(mn +mi

41.37mu

)−1

cm3 s−1 g−1,

(A2)where we have taken mn = 2.37mu (the mean molecularweight for our abundances, taking all the hydrogen to bemolecular), and mi = 39mu (K+, appropriate for thermalionization; Balbus & Terquem 2001)14. The diffusivity co-efficients for Ohmic, Hall, and ambipolar diffusion are asfollows (in the absence of dust; Balbus & Terquem 2001;Wardle 2007; Lesur et al. 2014):

ηO =c2me

4πe2

nn

ne〈σv〉ne (A3)

ηH =Bc

4πene=

√ρ

4π

vAc

nee(A4)

ηA =B2

4πγiρρi=

v2A

γiρi, (A5)

where vA = B/√

4πρ is the Alfven speed.By introducing the isothermal sound speed cs =√

kT/µmu we can examine the ratios

ηH

ηO=√

4πvA

cs

√ntot

nn

√kT

〈σv〉ne

e

cmen−1/2

n (A6)

= 1.14vA

cs

√ntot

nn

(munn

10−6 g cm−3

)−1/2

ηA

ηH=√

4πne

ni

vA

cs

e

γicmi√ρ

√kT

µmu(A7)

= 6.61×10−3 ne

ni

vA

csfM (ρ−6)−1/2

(T

3000 K

)1/2

fM ≡(

mi

39mu

)−1(mn+mi

41.37mu

)( µ

2.37

)−1/2

, (A8)

where ρ−6 = ρ/10−6 g cm−3, and fM = 1 for the abundances

13 Here we use Gaussian units, thus our presentation differers

by factors of√

4π compared to that of Lesur et al. (2014).14 Note that Blaes & Balbus (1994) adopted mi = 30mu, ap-propriate for the interstellar medium; while Sano et al. (2000)

adopted mi = 24mu (Mg+) appropriate for ionization in accre-

tion discs due to irradiation.

c© 2015 RAS, MNRAS 000, 1–18


−2.0

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

2.0

Hei

ght

/1.

13×

108

cm

ws0438

Rem

0 50 100 150 200

Time (Orbits)

−2.0

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

2.0

Hei

ght

/1.

13×

108

cm

Ha

100 101 102 103

Elsasser Number

Figure A1. Horizontally averaged Elsasser numbers for the

hottest lower branch simulation (ws0438). Lower Elsasser num-

ber corresponds to non-ideal MHD effects being more important.Rem (Eq. A9, top panel) corresponds to Ohmic dissipation and

Ha (Eq. A10, bottom panel) corresponds to the Hall term. Pre-

vious works (e.g. Hawley et al. 1996; Lesur & Longaretti 2007)suggest that Ohmic dissipation is important for Rem . 100; we

have therefore plotted a dotted black contour at Rem, Ha = 100.Non-ideal MHD effects are thus likely important for the majority

of this simulation.

and masses we assume here. From this we immediately con-clude that the Hall term will likely be important wheneverthe Ohmic dissipation is important, as we are typically talk-ing about densities and temperatures in this range. How-ever, ambipolar diffusion is negligible compared to both ofthe other two non-ideal effects.

To determine how important the Ohmic and Hall termsare, it is useful to examine the following dimensionless El-sasser numbers which can be computed from simulation pro-files:

Rem ≡v2

A

ΩηO(A9)

Ha ≡ v2A

ΩηH(A10)

For the hottest lower branch simulation (ws0438) we havecomputed the horizontally averaged Elsasser numbers (seeFigure A1). Previous works (e.g. Hawley et al. 1996; Lesur& Longaretti 2007) show that Ohmic dissipation is impor-tant when Rem is less than a few hundred and much ofws0438 has Elsasser numbers below 100 (the dotted blackcontour of Figure A1). The other lower branch simulationsare colder than ws0438 and are therefore more susceptibleto non-ideal effects, implying that non-ideal MHD effectsare important in dwarf nova quiescence. Future work shouldtherefore include Ohmic dissipation and the Hall term in

our simulations, but the ambipolar diffusion term is proba-bly not necessary.

APPENDIX B: SIMULATION PARAMETERS

The parameters for 3D MHD shearing-box simulationswhich appear in this paper but not in Hirose et al. (2014)are listed below in Table B1.

c© 2015 RAS, MNRAS 000, 1–18


Teff Σ α τtot Teff,0 Σ0 α0 β0 h0/108 Nx Ny Nz Lx/h0 Ly/h0 Lz/h0 Lz/hP tth tmax

R = 1.25×109 cm

8932 10.7 0.090 5876 8709 10.9 0.042 10 0.226 32 64 384 0.50 2.00 6.00 15.2 13.8 1899427 12.2 0.068 4910 7943 12.5 0.026 10 0.210 32 64 384 0.50 2.00 6.00 11.5 17.1 114

9698 14.7 0.055 5162 7943 14.9 0.020 10 0.221 32 64 384 0.50 2.00 6.00 10.9 19.7 118

17405 85.8 0.037 6985 16218 86.4 0.026 100 0.447 32 64 512 0.50 2.00 6.00 13.4 24.4 13319471 106 0.042 6798 16218 107 0.020 10 0.462 32 64 512 0.50 2.00 6.00 13.0 20.9 139

R = 4.13×109 cm

7546 36.7 0.134 17221 8709 37.2 0.067 10 1.55 32 64 384 0.50 2.00 6.00 22.7 11.0 165

8079 40.4 0.099 16800 9549 40.6 0.081 10 1.71 32 64 384 0.50 2.00 6.00 19.9 12.8 1148542 45.2 0.076 14139 9549 45.5 0.070 10 1.75 32 64 384 0.50 2.00 6.00 16.4 15.1 122

9324 52.0 0.069 9283 9549 52.5 0.058 10 1.80 32 64 384 0.50 2.00 6.00 13.9 15.9 123

Table B1. Parameters for simulation runs. Teff , Σ, α, and τtot are the time averaged effective temperature in K, column mass density

in g cm−2, Shakura & Sunyaev (1973) α-parameter, and optical depth respectively. Teff,0, Σ0, α0, and β0 are the initial condition values

where β0 is the initial ratio of gas to magnetic pressure. h0/108 is the simulation length unit over 108 cm. Nx, Ny , Nz are the numberof grid cells in the x, y, and z directions respectively. Lx/h0, Ly/h0, Lz/h0 are dimensions of the simulation domain in the x, y, and

z directions respectively. Lz/hP is the height of the simulation domain over the time averaged pressure scale height. tth is the time

averaged thermal time. tmax is the maximum runtime used for time averaging.

c© 2015 RAS, MNRAS 000, 1–18

Dwarf Nova Outbursts with Magnetorotational Turbulence

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